How do i change just one side of a mirrored model?
If you want to break symmetry you have to remove the mirror modifier by clicking "Apply". Unfortunately once i'ts done you can't go back, so be sure that your cat has enough details before you apply the modifier. 

Another way to do that to keep the mirror modifier would be to add another modifier on the stack, over the mirror modifier. In your case, you could add a boolean modifier to make the cut, but think to hide this modifier if you want to edit the base mesh if it is high poly, it can slow down blender.